Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 4.47·7-s + 9-s + 3.66·11-s + 3.81·13-s − 15-s − 6.96·17-s − 0.0738·19-s + 4.47·21-s − 7.13·23-s + 25-s + 27-s + 0.538·29-s + 2.70·31-s + 3.66·33-s − 4.47·35-s + 8.36·37-s + 3.81·39-s − 0.701·41-s + 10.9·43-s − 45-s + 2.26·47-s + 13.0·49-s − 6.96·51-s + 1.74·53-s − 3.66·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.69·7-s + 0.333·9-s + 1.10·11-s + 1.05·13-s − 0.258·15-s − 1.69·17-s − 0.0169·19-s + 0.976·21-s − 1.48·23-s + 0.200·25-s + 0.192·27-s + 0.100·29-s + 0.485·31-s + 0.637·33-s − 0.756·35-s + 1.37·37-s + 0.610·39-s − 0.109·41-s + 1.66·43-s − 0.149·45-s + 0.330·47-s + 1.86·49-s − 0.975·51-s + 0.240·53-s − 0.493·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.375075234$
$L(\frac12)$  $\approx$  $3.375075234$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
67 \( 1 - T \)
good7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 3.66T + 11T^{2} \)
13 \( 1 - 3.81T + 13T^{2} \)
17 \( 1 + 6.96T + 17T^{2} \)
19 \( 1 + 0.0738T + 19T^{2} \)
23 \( 1 + 7.13T + 23T^{2} \)
29 \( 1 - 0.538T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 - 8.36T + 37T^{2} \)
41 \( 1 + 0.701T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 2.26T + 47T^{2} \)
53 \( 1 - 1.74T + 53T^{2} \)
59 \( 1 - 0.807T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + 1.83T + 73T^{2} \)
79 \( 1 - 4.17T + 79T^{2} \)
83 \( 1 - 3.56T + 83T^{2} \)
89 \( 1 - 8.57T + 89T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.944084233399714378382231848734, −7.33900940384737994004210813888, −6.43291244478583211228671105688, −5.88821093828527551079755032990, −4.68818440818732259680056250993, −4.24581654069651171054702663388, −3.79076820916820763183438534601, −2.49355200325312525564979424923, −1.80310609603500606193762052574, −0.945971938827641472057716976106, 0.945971938827641472057716976106, 1.80310609603500606193762052574, 2.49355200325312525564979424923, 3.79076820916820763183438534601, 4.24581654069651171054702663388, 4.68818440818732259680056250993, 5.88821093828527551079755032990, 6.43291244478583211228671105688, 7.33900940384737994004210813888, 7.944084233399714378382231848734

Graph of the $Z$-function along the critical line